# Properties

 Label 33.4.e.c Level $33$ Weight $4$ Character orbit 33.e Analytic conductor $1.947$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.e (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 3 x^{11} + 21 x^{10} - 26 x^{9} + 281 x^{8} + 486 x^{7} + 3506 x^{6} + 15102 x^{5} + 46669 x^{4} + 41850 x^{3} + 16292 x^{2} + 616 x + 1936$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{2} -3 \beta_{6} q^{3} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{8} - \beta_{9} ) q^{4} + ( 4 + \beta_{1} + 4 \beta_{3} - \beta_{4} + 4 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{5} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{6} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{7} + ( 2 \beta_{1} + 14 \beta_{3} - 2 \beta_{4} + 14 \beta_{5} + 10 \beta_{6} + 3 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{8} + ( -9 - 9 \beta_{3} - 9 \beta_{5} - 9 \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{2} -3 \beta_{6} q^{3} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{8} - \beta_{9} ) q^{4} + ( 4 + \beta_{1} + 4 \beta_{3} - \beta_{4} + 4 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{5} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{6} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{7} + ( 2 \beta_{1} + 14 \beta_{3} - 2 \beta_{4} + 14 \beta_{5} + 10 \beta_{6} + 3 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{8} + ( -9 - 9 \beta_{3} - 9 \beta_{5} - 9 \beta_{6} ) q^{9} + ( 8 + 4 \beta_{1} - 10 \beta_{2} - 4 \beta_{7} + 4 \beta_{9} + 3 \beta_{11} ) q^{10} + ( -4 - 2 \beta_{1} + 6 \beta_{2} - \beta_{3} - 3 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - 4 \beta_{10} - 4 \beta_{11} ) q^{11} + ( -6 - 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{9} ) q^{12} + ( -14 + 9 \beta_{2} - 15 \beta_{3} + 9 \beta_{4} - 14 \beta_{5} - 15 \beta_{6} + 6 \beta_{7} - 9 \beta_{8} - 6 \beta_{10} - 6 \beta_{11} ) q^{13} + ( -6 \beta_{1} - 20 \beta_{3} + 6 \beta_{4} - 20 \beta_{5} - 14 \beta_{6} + \beta_{8} + 6 \beta_{9} - 6 \beta_{10} + \beta_{11} ) q^{14} + ( 12 - 3 \beta_{1} + 3 \beta_{2} + 12 \beta_{3} + 3 \beta_{4} + 6 \beta_{8} + 6 \beta_{9} - 3 \beta_{10} ) q^{15} + ( 34 + 7 \beta_{1} + 3 \beta_{3} + 10 \beta_{4} + 34 \beta_{6} + 10 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} ) q^{16} + ( -5 - \beta_{1} + 6 \beta_{3} + 14 \beta_{4} - 5 \beta_{6} + 14 \beta_{7} + 5 \beta_{8} + 5 \beta_{10} ) q^{17} + ( -9 \beta_{4} + 9 \beta_{5} ) q^{18} + ( 14 \beta_{1} - 9 \beta_{2} + 48 \beta_{3} - 14 \beta_{4} + 48 \beta_{5} + 5 \beta_{6} - 9 \beta_{7} - \beta_{8} + 5 \beta_{9} - 5 \beta_{10} - \beta_{11} ) q^{19} + ( -64 - 9 \beta_{2} - 87 \beta_{3} - 9 \beta_{4} - 64 \beta_{5} - 87 \beta_{6} - 25 \beta_{7} - 8 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{20} + ( -6 + 6 \beta_{1} - 3 \beta_{2} - 6 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 3 \beta_{9} - 6 \beta_{11} ) q^{21} + ( 31 + 19 \beta_{1} - 6 \beta_{2} + 62 \beta_{3} - 23 \beta_{4} + 42 \beta_{5} + 87 \beta_{6} - 3 \beta_{7} + \beta_{8} - 8 \beta_{9} + 10 \beta_{10} + 7 \beta_{11} ) q^{22} + ( -8 - 25 \beta_{1} - 4 \beta_{2} + 23 \beta_{5} + 23 \beta_{6} + 25 \beta_{7} - 4 \beta_{9} + 7 \beta_{11} ) q^{23} + ( -12 + 6 \beta_{2} + 30 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} + 30 \beta_{6} + 12 \beta_{8} + 9 \beta_{10} + 9 \beta_{11} ) q^{24} + ( -25 \beta_{1} + 12 \beta_{2} - 42 \beta_{3} + 25 \beta_{4} - 42 \beta_{5} + 46 \beta_{6} + 12 \beta_{7} - 4 \beta_{8} - 19 \beta_{9} + 19 \beta_{10} - 4 \beta_{11} ) q^{25} + ( 65 - 19 \beta_{1} + 19 \beta_{2} + 65 \beta_{3} - 14 \beta_{4} + 44 \beta_{5} - 6 \beta_{8} - 6 \beta_{9} + 3 \beta_{10} ) q^{26} -27 \beta_{3} q^{27} + ( -61 + \beta_{1} - 34 \beta_{3} - 15 \beta_{4} - 61 \beta_{6} - 15 \beta_{7} - 8 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} - 2 \beta_{11} ) q^{28} + ( 44 + 14 \beta_{1} - 14 \beta_{2} + 44 \beta_{3} - 11 \beta_{4} + 125 \beta_{5} + 3 \beta_{8} + 3 \beta_{9} + 6 \beta_{10} ) q^{29} + ( 18 \beta_{1} - 30 \beta_{2} - 18 \beta_{4} - 24 \beta_{6} - 30 \beta_{7} - 12 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} - 12 \beta_{11} ) q^{30} + ( -9 + 11 \beta_{2} - 57 \beta_{3} + 11 \beta_{4} - 9 \beta_{5} - 57 \beta_{6} - 36 \beta_{7} + 27 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{31} + ( 80 + 35 \beta_{1} - 9 \beta_{2} - 27 \beta_{5} - 27 \beta_{6} - 35 \beta_{7} - 11 \beta_{9} - 10 \beta_{11} ) q^{32} + ( 33 - 12 \beta_{1} + 27 \beta_{2} + 18 \beta_{3} + 12 \beta_{4} + 21 \beta_{5} + 30 \beta_{6} + 18 \beta_{7} - 6 \beta_{8} + 9 \beta_{9} - 12 \beta_{10} + 6 \beta_{11} ) q^{33} + ( 21 - \beta_{1} - 40 \beta_{2} - 149 \beta_{5} - 149 \beta_{6} + \beta_{7} + 23 \beta_{9} - 14 \beta_{11} ) q^{34} + ( -186 + 36 \beta_{2} - 53 \beta_{3} + 36 \beta_{4} - 186 \beta_{5} - 53 \beta_{6} + 9 \beta_{7} + \beta_{8} + 7 \beta_{10} + 7 \beta_{11} ) q^{35} + ( 9 \beta_{2} + 9 \beta_{3} + 9 \beta_{5} + 27 \beta_{6} + 9 \beta_{7} + 9 \beta_{8} + 9 \beta_{11} ) q^{36} + ( 103 - 39 \beta_{1} + 39 \beta_{2} + 103 \beta_{3} - 9 \beta_{4} + 72 \beta_{5} + 9 \beta_{10} ) q^{37} + ( 76 - 17 \beta_{1} - 99 \beta_{3} + 18 \beta_{4} + 76 \beta_{6} + 18 \beta_{7} - 16 \beta_{8} - 5 \beta_{9} - 11 \beta_{10} - 5 \beta_{11} ) q^{38} + ( -3 - 9 \beta_{1} - 45 \beta_{3} + 27 \beta_{4} - 3 \beta_{6} + 27 \beta_{7} - 27 \beta_{8} - 9 \beta_{9} - 18 \beta_{10} - 9 \beta_{11} ) q^{39} + ( -141 - 27 \beta_{1} + 27 \beta_{2} - 141 \beta_{3} - 3 \beta_{4} + 167 \beta_{5} + 3 \beta_{8} + 3 \beta_{9} - 45 \beta_{10} ) q^{40} + ( 43 \beta_{1} - 24 \beta_{2} + 147 \beta_{3} - 43 \beta_{4} + 147 \beta_{5} + 225 \beta_{6} - 24 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{41} + ( 18 - 18 \beta_{2} - 42 \beta_{3} - 18 \beta_{4} + 18 \beta_{5} - 42 \beta_{6} - 15 \beta_{8} + 3 \beta_{10} + 3 \beta_{11} ) q^{42} + ( -39 \beta_{1} - 26 \beta_{2} - 117 \beta_{5} - 117 \beta_{6} + 39 \beta_{7} + 19 \beta_{11} ) q^{43} + ( 112 + 27 \beta_{1} + 9 \beta_{2} + 92 \beta_{3} + 31 \beta_{4} - 20 \beta_{5} + 177 \beta_{6} + 21 \beta_{7} + 20 \beta_{8} + 20 \beta_{9} + 16 \beta_{10} + \beta_{11} ) q^{44} + ( -9 \beta_{1} - 36 \beta_{5} - 36 \beta_{6} + 9 \beta_{7} + 18 \beta_{9} + 9 \beta_{11} ) q^{45} + ( -210 + 19 \beta_{2} + 53 \beta_{3} + 19 \beta_{4} - 210 \beta_{5} + 53 \beta_{6} + 79 \beta_{7} + 43 \beta_{8} + 4 \beta_{10} + 4 \beta_{11} ) q^{46} + ( 73 \beta_{1} - 83 \beta_{2} - \beta_{3} - 73 \beta_{4} - \beta_{5} + 104 \beta_{6} - 83 \beta_{7} - 9 \beta_{8} + 26 \beta_{9} - 26 \beta_{10} - 9 \beta_{11} ) q^{47} + ( 102 + 30 \beta_{1} - 30 \beta_{2} + 102 \beta_{3} + 21 \beta_{4} + 93 \beta_{5} - 6 \beta_{8} - 6 \beta_{9} + 12 \beta_{10} ) q^{48} + ( -123 + 56 \beta_{1} - 203 \beta_{3} - 29 \beta_{4} - 123 \beta_{6} - 29 \beta_{7} + 6 \beta_{8} - \beta_{9} + 7 \beta_{10} - \beta_{11} ) q^{49} + ( -140 + 194 \beta_{3} + 36 \beta_{4} - 140 \beta_{6} + 36 \beta_{7} + 29 \beta_{8} + 25 \beta_{9} + 4 \beta_{10} + 25 \beta_{11} ) q^{50} + ( -15 + 42 \beta_{1} - 42 \beta_{2} - 15 \beta_{3} - 3 \beta_{4} - 33 \beta_{5} + 15 \beta_{8} + 15 \beta_{9} ) q^{51} + ( -49 \beta_{1} + 67 \beta_{2} + 146 \beta_{3} + 49 \beta_{4} + 146 \beta_{5} + 223 \beta_{6} + 67 \beta_{7} - 3 \beta_{8} + 11 \beta_{9} - 11 \beta_{10} - 3 \beta_{11} ) q^{52} + ( 83 + 9 \beta_{2} - 133 \beta_{3} + 9 \beta_{4} + 83 \beta_{5} - 133 \beta_{6} + 74 \beta_{7} - 15 \beta_{8} + 6 \beta_{10} + 6 \beta_{11} ) q^{53} + ( -27 + 27 \beta_{2} ) q^{54} + ( 378 - 43 \beta_{1} - 67 \beta_{2} + 19 \beta_{3} - 62 \beta_{4} + 136 \beta_{5} + 167 \beta_{6} - 39 \beta_{7} - 16 \beta_{8} - 25 \beta_{9} - 8 \beta_{10} - 23 \beta_{11} ) q^{55} + ( -37 - 24 \beta_{1} + 75 \beta_{2} + 70 \beta_{5} + 70 \beta_{6} + 24 \beta_{7} - 34 \beta_{9} - 29 \beta_{11} ) q^{56} + ( -129 + 15 \beta_{2} + 15 \beta_{3} + 15 \beta_{4} - 129 \beta_{5} + 15 \beta_{6} - 27 \beta_{7} - 18 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} ) q^{57} + ( -103 \beta_{1} + 49 \beta_{2} - 188 \beta_{3} + 103 \beta_{4} - 188 \beta_{5} + 7 \beta_{6} + 49 \beta_{7} - 9 \beta_{8} + 26 \beta_{9} - 26 \beta_{10} - 9 \beta_{11} ) q^{58} + ( 61 + 61 \beta_{3} + 76 \beta_{4} - 187 \beta_{5} - 14 \beta_{8} - 14 \beta_{9} - 54 \beta_{10} ) q^{59} + ( -69 - 48 \beta_{1} - 261 \beta_{3} - 27 \beta_{4} - 69 \beta_{6} - 27 \beta_{7} - 24 \beta_{8} - 18 \beta_{9} - 6 \beta_{10} - 18 \beta_{11} ) q^{60} + ( -86 + 26 \beta_{1} + 157 \beta_{3} - 123 \beta_{4} - 86 \beta_{6} - 123 \beta_{7} + 44 \beta_{8} + 5 \beta_{9} + 39 \beta_{10} + 5 \beta_{11} ) q^{61} + ( 46 - 18 \beta_{1} + 18 \beta_{2} + 46 \beta_{3} + 5 \beta_{4} + 369 \beta_{5} - 32 \beta_{8} - 32 \beta_{9} + 61 \beta_{10} ) q^{62} + ( -9 \beta_{1} - 9 \beta_{2} - 18 \beta_{3} + 9 \beta_{4} - 18 \beta_{5} - 9 \beta_{7} + 9 \beta_{8} + 18 \beta_{9} - 18 \beta_{10} + 9 \beta_{11} ) q^{63} + ( -138 + 60 \beta_{2} - 194 \beta_{3} + 60 \beta_{4} - 138 \beta_{5} - 194 \beta_{6} + 28 \beta_{7} - 26 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} ) q^{64} + ( 277 - 53 \beta_{1} + 37 \beta_{2} - 246 \beta_{5} - 246 \beta_{6} + 53 \beta_{7} + 29 \beta_{9} + 33 \beta_{11} ) q^{65} + ( 135 + 9 \beta_{1} + 51 \beta_{2} + 261 \beta_{3} + 39 \beta_{4} + 75 \beta_{5} + 168 \beta_{6} - 18 \beta_{7} + 27 \beta_{8} - 18 \beta_{9} + 21 \beta_{10} - 3 \beta_{11} ) q^{66} + ( -217 + 201 \beta_{1} - 69 \beta_{2} - 168 \beta_{5} - 168 \beta_{6} - 201 \beta_{7} - 63 \beta_{9} + 33 \beta_{11} ) q^{67} + ( -352 - 143 \beta_{2} - 655 \beta_{3} - 143 \beta_{4} - 352 \beta_{5} - 655 \beta_{6} - 171 \beta_{7} - 73 \beta_{8} - 46 \beta_{10} - 46 \beta_{11} ) q^{68} + ( 87 \beta_{1} - 12 \beta_{2} + 69 \beta_{3} - 87 \beta_{4} + 69 \beta_{5} + 93 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} - 21 \beta_{9} + 21 \beta_{10} + 12 \beta_{11} ) q^{69} + ( 481 + 95 \beta_{1} - 95 \beta_{2} + 481 \beta_{3} - 90 \beta_{4} + 398 \beta_{5} + 23 \beta_{8} + 23 \beta_{9} + 24 \beta_{10} ) q^{70} + ( 48 - 78 \beta_{1} - 101 \beta_{3} - 57 \beta_{4} + 48 \beta_{6} - 57 \beta_{7} + 40 \beta_{8} + \beta_{9} + 39 \beta_{10} + \beta_{11} ) q^{71} + ( 126 - 18 \beta_{1} + 90 \beta_{3} + 18 \beta_{4} + 126 \beta_{6} + 18 \beta_{7} + 36 \beta_{8} + 9 \beta_{9} + 27 \beta_{10} + 9 \beta_{11} ) q^{72} + ( -78 - 14 \beta_{1} + 14 \beta_{2} - 78 \beta_{3} + 73 \beta_{4} + 43 \beta_{5} - 29 \beta_{8} - 29 \beta_{9} + 94 \beta_{10} ) q^{73} + ( -120 \beta_{1} - \beta_{2} + 287 \beta_{3} + 120 \beta_{4} + 287 \beta_{5} + 461 \beta_{6} - \beta_{7} + 48 \beta_{8} - 21 \beta_{9} + 21 \beta_{10} + 48 \beta_{11} ) q^{74} + ( 264 - 39 \beta_{2} + 138 \beta_{3} - 39 \beta_{4} + 264 \beta_{5} + 138 \beta_{6} + 36 \beta_{7} + 45 \beta_{8} - 12 \beta_{10} - 12 \beta_{11} ) q^{75} + ( 13 - 7 \beta_{1} + 58 \beta_{2} + 183 \beta_{5} + 183 \beta_{6} + 7 \beta_{7} - 13 \beta_{9} - 48 \beta_{11} ) q^{76} + ( -95 - 81 \beta_{1} + 24 \beta_{2} - 408 \beta_{3} + 75 \beta_{4} - 232 \beta_{5} + 97 \beta_{6} + 18 \beta_{7} - 24 \beta_{8} - 13 \beta_{9} - 13 \beta_{10} - 4 \beta_{11} ) q^{77} + ( -132 - 57 \beta_{1} + 99 \beta_{2} - 195 \beta_{5} - 195 \beta_{6} + 57 \beta_{7} - 18 \beta_{9} - 9 \beta_{11} ) q^{78} + ( 97 - 193 \beta_{2} + 232 \beta_{3} - 193 \beta_{4} + 97 \beta_{5} + 232 \beta_{6} - 13 \beta_{7} - 64 \beta_{8} + 17 \beta_{10} + 17 \beta_{11} ) q^{79} + ( 36 \beta_{1} + 168 \beta_{2} + 266 \beta_{3} - 36 \beta_{4} + 266 \beta_{5} + 62 \beta_{6} + 168 \beta_{7} + 8 \beta_{8} - 69 \beta_{9} + 69 \beta_{10} + 8 \beta_{11} ) q^{80} + 81 \beta_{5} q^{81} + ( 301 + 144 \beta_{1} + 15 \beta_{3} + 109 \beta_{4} + 301 \beta_{6} + 109 \beta_{7} + \beta_{8} + 21 \beta_{9} - 20 \beta_{10} + 21 \beta_{11} ) q^{82} + ( -447 - 36 \beta_{1} + 94 \beta_{3} + 51 \beta_{4} - 447 \beta_{6} + 51 \beta_{7} - 44 \beta_{8} - 23 \beta_{9} - 21 \beta_{10} - 23 \beta_{11} ) q^{83} + ( -183 - 45 \beta_{1} + 45 \beta_{2} - 183 \beta_{3} + 3 \beta_{4} - 81 \beta_{5} - 18 \beta_{8} - 18 \beta_{9} - 6 \beta_{10} ) q^{84} + ( -131 \beta_{1} + 260 \beta_{2} + 119 \beta_{3} + 131 \beta_{4} + 119 \beta_{5} + 355 \beta_{6} + 260 \beta_{7} - 29 \beta_{9} + 29 \beta_{10} ) q^{85} + ( -508 - 125 \beta_{2} - 281 \beta_{3} - 125 \beta_{4} - 508 \beta_{5} - 281 \beta_{6} - 59 \beta_{7} + 51 \beta_{8} - 26 \beta_{10} - 26 \beta_{11} ) q^{86} + ( -375 + 42 \beta_{1} - 9 \beta_{2} - 132 \beta_{5} - 132 \beta_{6} - 42 \beta_{7} + 9 \beta_{9} - 18 \beta_{11} ) q^{87} + ( 127 + 82 \beta_{1} - 241 \beta_{2} + 232 \beta_{3} - 2 \beta_{4} - 104 \beta_{5} - 8 \beta_{6} - 126 \beta_{7} - 23 \beta_{8} + 88 \beta_{9} - 71 \beta_{10} + 18 \beta_{11} ) q^{88} + ( 140 + 27 \beta_{1} + 128 \beta_{2} + 297 \beta_{5} + 297 \beta_{6} - 27 \beta_{7} + 90 \beta_{9} - \beta_{11} ) q^{89} + ( -72 - 36 \beta_{2} - 72 \beta_{3} - 36 \beta_{4} - 72 \beta_{5} - 72 \beta_{6} - 90 \beta_{7} - 9 \beta_{8} - 36 \beta_{10} - 36 \beta_{11} ) q^{90} + ( -45 \beta_{1} - 56 \beta_{2} - 85 \beta_{3} + 45 \beta_{4} - 85 \beta_{5} + 115 \beta_{6} - 56 \beta_{7} + 34 \beta_{8} + 45 \beta_{9} - 45 \beta_{10} + 34 \beta_{11} ) q^{91} + ( 380 + 244 \beta_{1} - 244 \beta_{2} + 380 \beta_{3} - 85 \beta_{4} - 367 \beta_{5} + 55 \beta_{8} + 55 \beta_{9} + 41 \beta_{10} ) q^{92} + ( -144 - 141 \beta_{1} - 171 \beta_{3} + 33 \beta_{4} - 144 \beta_{6} + 33 \beta_{7} + 81 \beta_{8} + 75 \beta_{9} + 6 \beta_{10} + 75 \beta_{11} ) q^{93} + ( -227 + 193 \beta_{1} - 850 \beta_{3} - 161 \beta_{4} - 227 \beta_{6} - 161 \beta_{7} - 163 \beta_{8} - 62 \beta_{9} - 101 \beta_{10} - 62 \beta_{11} ) q^{94} + ( 231 + 28 \beta_{1} - 28 \beta_{2} + 231 \beta_{3} + 101 \beta_{4} + 281 \beta_{5} + 83 \beta_{8} + 83 \beta_{9} - 208 \beta_{10} ) q^{95} + ( -78 \beta_{1} - 27 \beta_{2} - 81 \beta_{3} + 78 \beta_{4} - 81 \beta_{5} - 321 \beta_{6} - 27 \beta_{7} + 33 \beta_{8} + 30 \beta_{9} - 30 \beta_{10} + 33 \beta_{11} ) q^{96} + ( 326 + 171 \beta_{2} + 178 \beta_{3} + 171 \beta_{4} + 326 \beta_{5} + 178 \beta_{6} + 314 \beta_{7} - 115 \beta_{8} + 6 \beta_{10} + 6 \beta_{11} ) q^{97} + ( 400 - 63 \beta_{1} + 205 \beta_{2} + 337 \beta_{5} + 337 \beta_{6} + 63 \beta_{7} + 41 \beta_{9} + 31 \beta_{11} ) q^{98} + ( 27 - 27 \beta_{1} + 45 \beta_{2} + 90 \beta_{3} + 45 \beta_{4} + 36 \beta_{5} - 9 \beta_{6} + 81 \beta_{7} - 45 \beta_{8} - 36 \beta_{9} + 18 \beta_{10} - 9 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 9q^{3} - 16q^{4} + 28q^{5} + 12q^{7} - 112q^{8} - 27q^{9} + O(q^{10})$$ $$12q + 9q^{3} - 16q^{4} + 28q^{5} + 12q^{7} - 112q^{8} - 27q^{9} + 100q^{10} - 54q^{11} - 102q^{12} - 18q^{13} + 156q^{14} + 111q^{15} + 308q^{16} - 80q^{17} - 45q^{18} - 280q^{19} - 15q^{20} - 6q^{21} - 193q^{22} - 392q^{23} - 264q^{24} + 77q^{25} + 406q^{26} + 81q^{27} - 429q^{28} + 13q^{29} + 120q^{30} + 413q^{31} + 1314q^{32} + 177q^{33} + 1060q^{34} - 1239q^{35} - 144q^{36} + 654q^{37} + 912q^{38} + 54q^{39} - 1803q^{40} - 1490q^{41} + 342q^{42} + 416q^{43} + 695q^{44} + 162q^{45} - 2369q^{46} - 150q^{47} + 711q^{48} - 301q^{49} - 1878q^{50} - 1661q^{52} + 1359q^{53} - 270q^{54} + 3300q^{55} - 858q^{56} - 1110q^{57} + 955q^{58} + 1262q^{59} + 45q^{60} - 1044q^{61} - 701q^{62} + 108q^{63} + 78q^{64} + 4556q^{65} + 369q^{66} - 528q^{67} + 703q^{68} - 594q^{69} + 3050q^{70} + 558q^{71} + 792q^{72} - 699q^{73} - 3224q^{74} + 1284q^{75} - 868q^{76} + 390q^{77} - 558q^{78} - 1252q^{79} - 1914q^{80} - 243q^{81} + 2987q^{82} - 4464q^{83} - 1443q^{84} - 2170q^{85} - 3209q^{86} - 3474q^{87} + 1302q^{88} + 316q^{89} - 90q^{90} + 176q^{91} + 4595q^{92} - 1239q^{93} + 1247q^{94} + 1466q^{95} + 1398q^{96} + 1608q^{97} + 2810q^{98} - 171q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 21 x^{10} - 26 x^{9} + 281 x^{8} + 486 x^{7} + 3506 x^{6} + 15102 x^{5} + 46669 x^{4} + 41850 x^{3} + 16292 x^{2} + 616 x + 1936$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-6986944582847976513 \nu^{11} + 23163569380150890824 \nu^{10} - 153931767460290190961 \nu^{9} + 235140221660543649730 \nu^{8} - 2069964535083095660825 \nu^{7} - 2495959479343529156658 \nu^{6} - 24989208660214669470346 \nu^{5} - 94664257433687411316282 \nu^{4} - 296324885413782065156597 \nu^{3} - 189062407549799154437530 \nu^{2} + 15961334394692549325760 \nu - 78558043089841259336898$$$$)/$$$$18\!\cdots\!89$$ $$\beta_{3}$$ $$=$$ $$($$$$16531513000088361634 \nu^{11} - 44963669399566691738 \nu^{10} + 348915737436588292851 \nu^{9} - 389366357308863855609 \nu^{8} + 4870370919031939910121 \nu^{7} + 8818141577887473112246 \nu^{6} + 64754961937344144101221 \nu^{5} + 269893071475894974094208 \nu^{4} + 894540942164531756734248 \nu^{3} + 1091817446741743200538922 \nu^{2} + 1086067427049355865922117 \nu + 114186350384130631498056$$$$)/$$$$37\!\cdots\!78$$ $$\beta_{4}$$ $$=$$ $$($$$$-46187384624125069357 \nu^{11} + 186960657265634921981 \nu^{10} - 1135246953413097226587 \nu^{9} + 2286911209821250340128 \nu^{8} - 14734418497538625039305 \nu^{7} - 7812521628419409462428 \nu^{6} - 146176049454948748163474 \nu^{5} - 526488943354278938492894 \nu^{4} - 1563782666952777918802773 \nu^{3} + 130778948334237759252744 \nu^{2} + 333828271431495750175728 \nu + 128217599677938392211744$$$$)/$$$$74\!\cdots\!56$$ $$\beta_{5}$$ $$=$$ $$($$$$126037762753186476125 \nu^{11} - 544907092573386951849 \nu^{10} + 3199007029650462175031 \nu^{9} - 6940741800865098547612 \nu^{8} + 40971492462543808505353 \nu^{7} + 12090494382928494599052 \nu^{6} + 379113767684947340266462 \nu^{5} + 1330219876312765163559874 \nu^{4} + 3579637355896770456213173 \nu^{3} - 1823378556747139397651160 \nu^{2} - 2464478201962668760690208 \nu - 1037472490797173716634624$$$$)/$$$$14\!\cdots\!12$$ $$\beta_{6}$$ $$=$$ $$($$$$66228099007199582754 \nu^{11} - 152496912397473678905 \nu^{10} + 1203829421885556315853 \nu^{9} - 586683620774091925017 \nu^{8} + 16323184611201832413746 \nu^{7} + 46921274615037622257749 \nu^{6} + 240008236747661146597952 \nu^{5} + 1146352800661676846914382 \nu^{4} + 3617288095921276266039320 \nu^{3} + 4335428610404080457057673 \nu^{2} + 948209240691057842975424 \nu - 293031762443060807199264$$$$)/$$$$74\!\cdots\!56$$ $$\beta_{7}$$ $$=$$ $$($$$$83396902156913761737 \nu^{11} - 276107005916773088203 \nu^{10} + 1831879984641125084181 \nu^{9} - 2777440564449204357114 \nu^{8} + 24581929157560066398849 \nu^{7} + 31387314263862222513894 \nu^{6} + 286601208392928499439938 \nu^{5} + 1151209497015844599032226 \nu^{4} + 3549029463983996711741205 \nu^{3} + 2258942716368791414859354 \nu^{2} + 557555876326568292963812 \nu + 122004554345084508889000$$$$)/$$$$74\!\cdots\!56$$ $$\beta_{8}$$ $$=$$ $$($$$$115983252017734326873 \nu^{11} - 907731642419289179416 \nu^{10} + 4260581331073491155112 \nu^{9} - 14982870263321782726327 \nu^{8} + 48978231317019716063651 \nu^{7} - 96960610820782954517143 \nu^{6} + 153626675667707795867104 \nu^{5} - 37492592919423453345032 \nu^{4} - 2612100325686493587623925 \nu^{3} - 18180721886271700252456903 \nu^{2} - 12600118248645652827302862 \nu + 1254998591973740725948584$$$$)/$$$$74\!\cdots\!56$$ $$\beta_{9}$$ $$=$$ $$($$$$533015920636260583195 \nu^{11} - 1748513562934959788065 \nu^{10} + 11720122199125039606691 \nu^{9} - 17844911247530639011218 \nu^{8} + 157628596649528102797719 \nu^{7} + 200252009113898715815570 \nu^{6} + 1848747365963578049544906 \nu^{5} + 7368699656691330615509042 \nu^{4} + 22745306836813671960695191 \nu^{3} + 14480156264424739008193918 \nu^{2} - 1216349985487171824694944 \nu - 13878174711647099254746312$$$$)/$$$$14\!\cdots\!12$$ $$\beta_{10}$$ $$=$$ $$($$$$-558862071085917679827 \nu^{11} + 2077472016341165596055 \nu^{10} - 13353277182944826227147 \nu^{9} + 24273942814112362823198 \nu^{8} - 176155127320564137754001 \nu^{7} - 147926364176448287773920 \nu^{6} - 1872569628726162495698116 \nu^{5} - 7250437516214319138907242 \nu^{4} - 21294618303526191256564867 \nu^{3} - 10899947865824080520193532 \nu^{2} - 9169360122713995076075770 \nu - 4297246498670572018907412$$$$)/$$$$74\!\cdots\!56$$ $$\beta_{11}$$ $$=$$ $$($$$$-1822543589778456797545 \nu^{11} + 5971100475229906293843 \nu^{10} - 39301103063307384239393 \nu^{9} + 55671505655896899767166 \nu^{8} - 511206662952446781194181 \nu^{7} - 780310465174366349243918 \nu^{6} - 5956967152038617789901382 \nu^{5} - 25822713828465960937971662 \nu^{4} - 75807033214760810401670173 \nu^{3} - 47868070482912537731555626 \nu^{2} + 3945481256433326717417376 \nu + 4059780316857578394660304$$$$)/$$$$14\!\cdots\!12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{8} - \beta_{7} + 10 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{10} - 16 \beta_{5} - 17 \beta_{4} - 11 \beta_{3} - \beta_{2} + \beta_{1} - 11$$ $$\nu^{4}$$ $$=$$ $$-16 \beta_{11} - 16 \beta_{10} - 16 \beta_{8} + 21 \beta_{7} - 162 \beta_{6} - 154 \beta_{5} - 32 \beta_{4} - 162 \beta_{3} - 32 \beta_{2} - 154$$ $$\nu^{5}$$ $$=$$ $$-32 \beta_{11} - 5 \beta_{9} + 61 \beta_{7} - 315 \beta_{6} - 315 \beta_{5} - 279 \beta_{2} - 61 \beta_{1} - 429$$ $$\nu^{6}$$ $$=$$ $$-29 \beta_{11} + 274 \beta_{10} - 29 \beta_{9} + 245 \beta_{8} + 807 \beta_{7} - 253 \beta_{6} + 807 \beta_{4} + 2805 \beta_{3} - 1249 \beta_{1} - 253$$ $$\nu^{7}$$ $$=$$ $$-168 \beta_{11} + 778 \beta_{10} - 778 \beta_{9} - 168 \beta_{8} + 5121 \beta_{7} - 2604 \beta_{6} + 7614 \beta_{5} + 7243 \beta_{4} + 7614 \beta_{3} + 5121 \beta_{2} - 7243 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$1344 \beta_{10} - 4953 \beta_{9} - 4953 \beta_{8} + 64453 \beta_{5} + 28886 \beta_{4} + 13703 \beta_{3} + 19145 \beta_{2} - 19145 \beta_{1} + 13703$$ $$\nu^{9}$$ $$=$$ $$4788 \beta_{11} + 4788 \beta_{10} - 13013 \beta_{8} - 98104 \beta_{7} + 62485 \beta_{6} + 238564 \beta_{5} + 60813 \beta_{4} + 62485 \beta_{3} + 60813 \beta_{2} + 238564$$ $$\nu^{10}$$ $$=$$ $$43012 \beta_{11} + 93316 \beta_{9} - 444424 \beta_{7} + 445776 \beta_{6} + 445776 \beta_{5} + 222753 \beta_{2} + 444424 \beta_{1} + 1400254$$ $$\nu^{11}$$ $$=$$ $$401412 \beta_{11} - 129437 \beta_{10} + 401412 \beta_{9} + 271975 \beta_{8} - 1593097 \beta_{7} + 4013999 \beta_{6} - 1593097 \beta_{4} - 1538878 \beta_{3} + 3545004 \beta_{1} + 4013999$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 −1.92306 + 1.39719i −0.654357 + 0.475418i 3.88644 − 2.82366i −1.17727 + 3.62326i 0.0936861 − 0.288336i 1.27457 − 3.92271i −1.92306 − 1.39719i −0.654357 − 0.475418i 3.88644 + 2.82366i −1.17727 − 3.62326i 0.0936861 + 0.288336i 1.27457 + 3.92271i
−2.73208 1.98497i −0.927051 + 2.85317i 1.05201 + 3.23775i −11.3314 + 8.23272i 8.19624 5.95492i 7.21292 + 22.1991i −4.79582 + 14.7600i −7.28115 5.29007i 47.2999
4.2 −1.46337 1.06320i −0.927051 + 2.85317i −1.46107 4.49672i 17.3626 12.6147i 4.39012 3.18961i −5.78498 17.8043i −7.11451 + 21.8962i −7.28115 5.29007i −38.8200
4.3 3.07742 + 2.23588i −0.927051 + 2.85317i 1.99923 + 6.15301i 2.08679 1.51614i −9.23226 + 6.70763i 0.454025 + 1.39734i 1.79887 5.53636i −7.28115 5.29007i 9.81184
16.1 −0.868251 2.67220i 2.42705 + 1.76336i 0.0853305 0.0619962i 3.76992 11.6026i 2.60475 8.01661i −5.42571 + 3.94201i −18.4246 13.3863i 2.78115 + 8.55951i −34.2777
16.2 0.402703 + 1.23939i 2.42705 + 1.76336i 5.09821 3.70407i −2.50364 + 7.70540i −1.20811 + 3.71818i −0.439746 + 0.319494i 15.0782 + 10.9549i 2.78115 + 8.55951i −10.5582
16.3 1.58358 + 4.87376i 2.42705 + 1.76336i −14.7737 + 10.7337i 4.61569 14.2056i −4.75075 + 14.6213i 9.98349 7.25343i −42.5421 30.9086i 2.78115 + 8.55951i 76.5442
25.1 −2.73208 + 1.98497i −0.927051 2.85317i 1.05201 3.23775i −11.3314 8.23272i 8.19624 + 5.95492i 7.21292 22.1991i −4.79582 14.7600i −7.28115 + 5.29007i 47.2999
25.2 −1.46337 + 1.06320i −0.927051 2.85317i −1.46107 + 4.49672i 17.3626 + 12.6147i 4.39012 + 3.18961i −5.78498 + 17.8043i −7.11451 21.8962i −7.28115 + 5.29007i −38.8200
25.3 3.07742 2.23588i −0.927051 2.85317i 1.99923 6.15301i 2.08679 + 1.51614i −9.23226 6.70763i 0.454025 1.39734i 1.79887 + 5.53636i −7.28115 + 5.29007i 9.81184
31.1 −0.868251 + 2.67220i 2.42705 1.76336i 0.0853305 + 0.0619962i 3.76992 + 11.6026i 2.60475 + 8.01661i −5.42571 3.94201i −18.4246 + 13.3863i 2.78115 8.55951i −34.2777
31.2 0.402703 1.23939i 2.42705 1.76336i 5.09821 + 3.70407i −2.50364 7.70540i −1.20811 3.71818i −0.439746 0.319494i 15.0782 10.9549i 2.78115 8.55951i −10.5582
31.3 1.58358 4.87376i 2.42705 1.76336i −14.7737 10.7337i 4.61569 + 14.2056i −4.75075 14.6213i 9.98349 + 7.25343i −42.5421 + 30.9086i 2.78115 8.55951i 76.5442
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.4.e.c 12
3.b odd 2 1 99.4.f.d 12
11.c even 5 1 inner 33.4.e.c 12
11.c even 5 1 363.4.a.v 6
11.d odd 10 1 363.4.a.u 6
33.f even 10 1 1089.4.a.bk 6
33.h odd 10 1 99.4.f.d 12
33.h odd 10 1 1089.4.a.bi 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.c 12 1.a even 1 1 trivial
33.4.e.c 12 11.c even 5 1 inner
99.4.f.d 12 3.b odd 2 1
99.4.f.d 12 33.h odd 10 1
363.4.a.u 6 11.d odd 10 1
363.4.a.v 6 11.c even 5 1
1089.4.a.bi 6 33.h odd 10 1
1089.4.a.bk 6 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T^{2} + 24 T^{3} - 71 T^{4} - 282 T^{5} + 991 T^{6} - 2439 T^{7} - 7897 T^{8} + 33786 T^{9} - 40364 T^{10} - 95640 T^{11} + 1189232 T^{12} - 765120 T^{13} - 2583296 T^{14} + 17298432 T^{15} - 32346112 T^{16} - 79921152 T^{17} + 259784704 T^{18} - 591396864 T^{19} - 1191182336 T^{20} + 3221225472 T^{21} - 4294967296 T^{22} + 68719476736 T^{24}$$
$3$ $$( 1 - 3 T + 9 T^{2} - 27 T^{3} + 81 T^{4} )^{3}$$
$5$ $$1 - 28 T + 166 T^{2} + 2008 T^{3} - 6243 T^{4} - 198220 T^{5} - 5895180 T^{6} + 127925504 T^{7} - 250932132 T^{8} - 5175713236 T^{9} - 4736493310 T^{10} - 636606527596 T^{11} + 19293489775396 T^{12} - 79575815949500 T^{13} - 74007707968750 T^{14} - 10108814914062500 T^{15} - 61262727539062500 T^{16} + 3903976562500000000 T^{17} - 22488327026367187500 T^{18} - 94518661499023437500 T^{19} -$$$$37\!\cdots\!75$$$$T^{20} +$$$$14\!\cdots\!00$$$$T^{21} +$$$$15\!\cdots\!50$$$$T^{22} -$$$$32\!\cdots\!00$$$$T^{23} +$$$$14\!\cdots\!25$$$$T^{24}$$
$7$ $$1 - 12 T - 292 T^{2} + 3268 T^{3} + 253084 T^{4} - 1719060 T^{5} - 106627353 T^{6} + 798814860 T^{7} + 47875350316 T^{8} - 186188964726 T^{9} - 14936850681692 T^{10} - 2595426222344 T^{11} + 7067195488099391 T^{12} - 890231194263992 T^{13} - 1757305545850382108 T^{14} - 7513396310289866682 T^{15} +$$$$66\!\cdots\!16$$$$T^{16} +$$$$37\!\cdots\!80$$$$T^{17} -$$$$17\!\cdots\!97$$$$T^{18} -$$$$96\!\cdots\!20$$$$T^{19} +$$$$48\!\cdots\!84$$$$T^{20} +$$$$21\!\cdots\!24$$$$T^{21} -$$$$65\!\cdots\!08$$$$T^{22} -$$$$92\!\cdots\!84$$$$T^{23} +$$$$26\!\cdots\!01$$$$T^{24}$$
$11$ $$1 + 54 T + 7 T^{2} - 126342 T^{3} - 3232702 T^{4} + 71168328 T^{5} + 7468757428 T^{6} + 94725044568 T^{7} - 5726928787822 T^{8} - 297907827176322 T^{9} + 21968998637047 T^{10} + 225571401148445154 T^{11} + 5559917313492231481 T^{12}$$
$13$ $$1 + 18 T + 2580 T^{2} + 28134 T^{3} + 9792792 T^{4} - 132508020 T^{5} + 18118966899 T^{6} - 845007146202 T^{7} + 41552489371800 T^{8} - 3032707536653446 T^{9} + 27211290556502316 T^{10} - 10062545621894725254 T^{11} + 99326497359265182237 T^{12} -$$$$22\!\cdots\!38$$$$T^{13} +$$$$13\!\cdots\!44$$$$T^{14} -$$$$32\!\cdots\!58$$$$T^{15} +$$$$96\!\cdots\!00$$$$T^{16} -$$$$43\!\cdots\!14$$$$T^{17} +$$$$20\!\cdots\!71$$$$T^{18} -$$$$32\!\cdots\!60$$$$T^{19} +$$$$53\!\cdots\!12$$$$T^{20} +$$$$33\!\cdots\!78$$$$T^{21} +$$$$67\!\cdots\!20$$$$T^{22} +$$$$10\!\cdots\!54$$$$T^{23} +$$$$12\!\cdots\!41$$$$T^{24}$$
$17$ $$1 + 80 T + 5981 T^{2} + 1271624 T^{3} + 115253234 T^{4} + 10488840088 T^{5} + 1050907094986 T^{6} + 88631392584416 T^{7} + 7529874686088563 T^{8} + 593783963038148776 T^{9} + 47965315498679770506 T^{10} +$$$$35\!\cdots\!00$$$$T^{11} +$$$$23\!\cdots\!87$$$$T^{12} +$$$$17\!\cdots\!00$$$$T^{13} +$$$$11\!\cdots\!14$$$$T^{14} +$$$$70\!\cdots\!72$$$$T^{15} +$$$$43\!\cdots\!43$$$$T^{16} +$$$$25\!\cdots\!88$$$$T^{17} +$$$$14\!\cdots\!74$$$$T^{18} +$$$$72\!\cdots\!96$$$$T^{19} +$$$$39\!\cdots\!14$$$$T^{20} +$$$$21\!\cdots\!52$$$$T^{21} +$$$$49\!\cdots\!69$$$$T^{22} +$$$$32\!\cdots\!60$$$$T^{23} +$$$$19\!\cdots\!81$$$$T^{24}$$
$19$ $$1 + 280 T + 35745 T^{2} + 3669090 T^{3} + 447314010 T^{4} + 44109561030 T^{5} + 2823999409126 T^{6} + 165028979914510 T^{7} + 11066477722204575 T^{8} - 31812268720570 T^{9} -$$$$10\!\cdots\!30$$$$T^{10} -$$$$10\!\cdots\!90$$$$T^{11} -$$$$79\!\cdots\!69$$$$T^{12} -$$$$72\!\cdots\!10$$$$T^{13} -$$$$47\!\cdots\!30$$$$T^{14} -$$$$10\!\cdots\!30$$$$T^{15} +$$$$24\!\cdots\!75$$$$T^{16} +$$$$25\!\cdots\!90$$$$T^{17} +$$$$29\!\cdots\!66$$$$T^{18} +$$$$31\!\cdots\!70$$$$T^{19} +$$$$21\!\cdots\!10$$$$T^{20} +$$$$12\!\cdots\!10$$$$T^{21} +$$$$82\!\cdots\!45$$$$T^{22} +$$$$44\!\cdots\!20$$$$T^{23} +$$$$10\!\cdots\!81$$$$T^{24}$$
$23$ $$( 1 + 196 T + 50798 T^{2} + 7868182 T^{3} + 1226347462 T^{4} + 145519235908 T^{5} + 18400674137554 T^{6} + 1770532543292636 T^{7} + 181543436760063718 T^{8} + 14171796950175270266 T^{9} +$$$$11\!\cdots\!58$$$$T^{10} +$$$$52\!\cdots\!72$$$$T^{11} +$$$$32\!\cdots\!69$$$$T^{12} )^{2}$$
$29$ $$1 - 13 T - 22361 T^{2} - 7268261 T^{3} + 1074887025 T^{4} - 8519956096 T^{5} + 9520176357252 T^{6} - 4111853450195056 T^{7} + 1165191373084224204 T^{8} -$$$$11\!\cdots\!16$$$$T^{9} -$$$$92\!\cdots\!90$$$$T^{10} -$$$$28\!\cdots\!98$$$$T^{11} +$$$$10\!\cdots\!18$$$$T^{12} -$$$$70\!\cdots\!22$$$$T^{13} -$$$$55\!\cdots\!90$$$$T^{14} -$$$$17\!\cdots\!04$$$$T^{15} +$$$$41\!\cdots\!64$$$$T^{16} -$$$$35\!\cdots\!44$$$$T^{17} +$$$$20\!\cdots\!72$$$$T^{18} -$$$$43\!\cdots\!84$$$$T^{19} +$$$$13\!\cdots\!25$$$$T^{20} -$$$$22\!\cdots\!49$$$$T^{21} -$$$$16\!\cdots\!61$$$$T^{22} -$$$$23\!\cdots\!57$$$$T^{23} +$$$$44\!\cdots\!21$$$$T^{24}$$
$31$ $$1 - 413 T + 23883 T^{2} + 9132189 T^{3} - 2225395551 T^{4} + 772309998732 T^{5} - 105670038412940 T^{6} - 21115198783619612 T^{7} + 5412938378503579212 T^{8} -$$$$44\!\cdots\!72$$$$T^{9} +$$$$77\!\cdots\!30$$$$T^{10} +$$$$13\!\cdots\!22$$$$T^{11} -$$$$73\!\cdots\!62$$$$T^{12} +$$$$39\!\cdots\!02$$$$T^{13} +$$$$68\!\cdots\!30$$$$T^{14} -$$$$11\!\cdots\!12$$$$T^{15} +$$$$42\!\cdots\!32$$$$T^{16} -$$$$49\!\cdots\!12$$$$T^{17} -$$$$73\!\cdots\!40$$$$T^{18} +$$$$16\!\cdots\!92$$$$T^{19} -$$$$13\!\cdots\!71$$$$T^{20} +$$$$16\!\cdots\!79$$$$T^{21} +$$$$13\!\cdots\!83$$$$T^{22} -$$$$67\!\cdots\!83$$$$T^{23} +$$$$48\!\cdots\!81$$$$T^{24}$$
$37$ $$1 - 654 T + 74760 T^{2} + 21496536 T^{3} + 3172278300 T^{4} - 4613962451016 T^{5} + 914242818447663 T^{6} + 42769681130442984 T^{7} - 18021863201800227252 T^{8} -$$$$13\!\cdots\!24$$$$T^{9} +$$$$48\!\cdots\!04$$$$T^{10} -$$$$23\!\cdots\!54$$$$T^{11} -$$$$91\!\cdots\!35$$$$T^{12} -$$$$12\!\cdots\!62$$$$T^{13} +$$$$12\!\cdots\!36$$$$T^{14} -$$$$17\!\cdots\!48$$$$T^{15} -$$$$11\!\cdots\!12$$$$T^{16} +$$$$14\!\cdots\!12$$$$T^{17} +$$$$15\!\cdots\!27$$$$T^{18} -$$$$39\!\cdots\!92$$$$T^{19} +$$$$13\!\cdots\!00$$$$T^{20} +$$$$47\!\cdots\!88$$$$T^{21} +$$$$83\!\cdots\!40$$$$T^{22} -$$$$36\!\cdots\!38$$$$T^{23} +$$$$28\!\cdots\!41$$$$T^{24}$$
$41$ $$1 + 1490 T + 880858 T^{2} + 231230716 T^{3} - 1247845038 T^{4} - 21538077143224 T^{5} - 7213863330589185 T^{6} - 376292589237375262 T^{7} +$$$$59\!\cdots\!24$$$$T^{8} +$$$$23\!\cdots\!26$$$$T^{9} +$$$$25\!\cdots\!30$$$$T^{10} -$$$$93\!\cdots\!70$$$$T^{11} -$$$$44\!\cdots\!67$$$$T^{12} -$$$$64\!\cdots\!70$$$$T^{13} +$$$$11\!\cdots\!30$$$$T^{14} +$$$$76\!\cdots\!86$$$$T^{15} +$$$$13\!\cdots\!44$$$$T^{16} -$$$$58\!\cdots\!62$$$$T^{17} -$$$$77\!\cdots\!85$$$$T^{18} -$$$$15\!\cdots\!84$$$$T^{19} -$$$$63\!\cdots\!18$$$$T^{20} +$$$$81\!\cdots\!96$$$$T^{21} +$$$$21\!\cdots\!58$$$$T^{22} +$$$$24\!\cdots\!90$$$$T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$43$ $$( 1 - 208 T + 292746 T^{2} - 28880738 T^{3} + 37781127122 T^{4} - 1726024887592 T^{5} + 3403809232886930 T^{6} - 137231060737777144 T^{7} +$$$$23\!\cdots\!78$$$$T^{8} -$$$$14\!\cdots\!34$$$$T^{9} +$$$$11\!\cdots\!46$$$$T^{10} -$$$$66\!\cdots\!56$$$$T^{11} +$$$$25\!\cdots\!49$$$$T^{12} )^{2}$$
$47$ $$1 + 150 T - 229834 T^{2} - 24484236 T^{3} + 4788515779 T^{4} - 1166873288142 T^{5} + 4483635535981786 T^{6} + 613830793032007656 T^{7} -$$$$52\!\cdots\!02$$$$T^{8} -$$$$66\!\cdots\!34$$$$T^{9} -$$$$21\!\cdots\!84$$$$T^{10} +$$$$28\!\cdots\!50$$$$T^{11} +$$$$78\!\cdots\!12$$$$T^{12} +$$$$29\!\cdots\!50$$$$T^{13} -$$$$22\!\cdots\!36$$$$T^{14} -$$$$74\!\cdots\!78$$$$T^{15} -$$$$60\!\cdots\!82$$$$T^{16} +$$$$74\!\cdots\!08$$$$T^{17} +$$$$56\!\cdots\!54$$$$T^{18} -$$$$15\!\cdots\!74$$$$T^{19} +$$$$64\!\cdots\!99$$$$T^{20} -$$$$34\!\cdots\!68$$$$T^{21} -$$$$33\!\cdots\!66$$$$T^{22} +$$$$22\!\cdots\!50$$$$T^{23} +$$$$15\!\cdots\!21$$$$T^{24}$$
$53$ $$1 - 1359 T + 701543 T^{2} - 211527147 T^{3} + 104475729361 T^{4} - 64065057259272 T^{5} + 21267785855167432 T^{6} - 4359429544881114288 T^{7} +$$$$22\!\cdots\!92$$$$T^{8} -$$$$13\!\cdots\!96$$$$T^{9} +$$$$51\!\cdots\!14$$$$T^{10} -$$$$19\!\cdots\!90$$$$T^{11} +$$$$79\!\cdots\!70$$$$T^{12} -$$$$28\!\cdots\!30$$$$T^{13} +$$$$11\!\cdots\!06$$$$T^{14} -$$$$44\!\cdots\!68$$$$T^{15} +$$$$10\!\cdots\!72$$$$T^{16} -$$$$31\!\cdots\!16$$$$T^{17} +$$$$23\!\cdots\!48$$$$T^{18} -$$$$10\!\cdots\!16$$$$T^{19} +$$$$25\!\cdots\!41$$$$T^{20} -$$$$76\!\cdots\!39$$$$T^{21} +$$$$37\!\cdots\!07$$$$T^{22} -$$$$10\!\cdots\!07$$$$T^{23} +$$$$11\!\cdots\!21$$$$T^{24}$$
$59$ $$1 - 1262 T + 275411 T^{2} + 319635658 T^{3} - 142409552824 T^{4} - 89740709300188 T^{5} + 73592642874959518 T^{6} + 1626297918888671692 T^{7} -$$$$12\!\cdots\!87$$$$T^{8} -$$$$11\!\cdots\!48$$$$T^{9} +$$$$33\!\cdots\!74$$$$T^{10} -$$$$27\!\cdots\!40$$$$T^{11} -$$$$45\!\cdots\!91$$$$T^{12} -$$$$57\!\cdots\!60$$$$T^{13} +$$$$14\!\cdots\!34$$$$T^{14} -$$$$10\!\cdots\!72$$$$T^{15} -$$$$21\!\cdots\!47$$$$T^{16} +$$$$59\!\cdots\!08$$$$T^{17} +$$$$55\!\cdots\!78$$$$T^{18} -$$$$13\!\cdots\!92$$$$T^{19} -$$$$45\!\cdots\!64$$$$T^{20} +$$$$20\!\cdots\!02$$$$T^{21} +$$$$36\!\cdots\!11$$$$T^{22} -$$$$34\!\cdots\!98$$$$T^{23} +$$$$56\!\cdots\!41$$$$T^{24}$$
$61$ $$1 + 1044 T + 504986 T^{2} + 424821958 T^{3} + 336901340533 T^{4} + 216760940872116 T^{5} + 139959802763217936 T^{6} + 77515379143789767996 T^{7} +$$$$43\!\cdots\!44$$$$T^{8} +$$$$24\!\cdots\!24$$$$T^{9} +$$$$13\!\cdots\!26$$$$T^{10} +$$$$64\!\cdots\!58$$$$T^{11} +$$$$30\!\cdots\!16$$$$T^{12} +$$$$14\!\cdots\!98$$$$T^{13} +$$$$67\!\cdots\!86$$$$T^{14} +$$$$28\!\cdots\!84$$$$T^{15} +$$$$11\!\cdots\!24$$$$T^{16} +$$$$46\!\cdots\!96$$$$T^{17} +$$$$19\!\cdots\!16$$$$T^{18} +$$$$67\!\cdots\!76$$$$T^{19} +$$$$23\!\cdots\!53$$$$T^{20} +$$$$67\!\cdots\!18$$$$T^{21} +$$$$18\!\cdots\!86$$$$T^{22} +$$$$86\!\cdots\!64$$$$T^{23} +$$$$18\!\cdots\!61$$$$T^{24}$$
$67$ $$( 1 + 264 T - 23619 T^{2} - 79408030 T^{3} + 61162456014 T^{4} + 20942991876474 T^{5} + 29398858296591696 T^{6} + 6298877065743949662 T^{7} +$$$$55\!\cdots\!66$$$$T^{8} -$$$$21\!\cdots\!10$$$$T^{9} -$$$$19\!\cdots\!59$$$$T^{10} +$$$$64\!\cdots\!52$$$$T^{11} +$$$$74\!\cdots\!09$$$$T^{12} )^{2}$$
$71$ $$1 - 558 T - 214976 T^{2} + 306079236 T^{3} + 184582139233 T^{4} - 28441797589062 T^{5} - 159756033892200858 T^{6} + 77696498899965080124 T^{7} +$$$$51\!\cdots\!50$$$$T^{8} -$$$$94\!\cdots\!34$$$$T^{9} -$$$$18\!\cdots\!78$$$$T^{10} -$$$$22\!\cdots\!30$$$$T^{11} +$$$$12\!\cdots\!84$$$$T^{12} -$$$$79\!\cdots\!30$$$$T^{13} -$$$$23\!\cdots\!38$$$$T^{14} -$$$$43\!\cdots\!54$$$$T^{15} +$$$$84\!\cdots\!50$$$$T^{16} +$$$$45\!\cdots\!24$$$$T^{17} -$$$$33\!\cdots\!38$$$$T^{18} -$$$$21\!\cdots\!02$$$$T^{19} +$$$$49\!\cdots\!73$$$$T^{20} +$$$$29\!\cdots\!76$$$$T^{21} -$$$$74\!\cdots\!76$$$$T^{22} -$$$$68\!\cdots\!38$$$$T^{23} +$$$$44\!\cdots\!21$$$$T^{24}$$
$73$ $$1 + 699 T + 552765 T^{2} - 109264537 T^{3} + 81592008873 T^{4} - 205792245455572 T^{5} - 26056419673978324 T^{6} - 97921969488226725576 T^{7} +$$$$54\!\cdots\!68$$$$T^{8} -$$$$20\!\cdots\!12$$$$T^{9} +$$$$18\!\cdots\!70$$$$T^{10} -$$$$13\!\cdots\!18$$$$T^{11} +$$$$93\!\cdots\!74$$$$T^{12} -$$$$53\!\cdots\!06$$$$T^{13} +$$$$27\!\cdots\!30$$$$T^{14} -$$$$12\!\cdots\!56$$$$T^{15} +$$$$12\!\cdots\!28$$$$T^{16} -$$$$87\!\cdots\!32$$$$T^{17} -$$$$90\!\cdots\!56$$$$T^{18} -$$$$27\!\cdots\!56$$$$T^{19} +$$$$42\!\cdots\!93$$$$T^{20} -$$$$22\!\cdots\!89$$$$T^{21} +$$$$43\!\cdots\!85$$$$T^{22} +$$$$21\!\cdots\!67$$$$T^{23} +$$$$12\!\cdots\!61$$$$T^{24}$$
$79$ $$1 + 1252 T - 706116 T^{2} - 920771466 T^{3} + 605427548280 T^{4} + 160463564258004 T^{5} - 635347097378782703 T^{6} - 61745950226810786696 T^{7} +$$$$19\!\cdots\!84$$$$T^{8} -$$$$60\!\cdots\!66$$$$T^{9} -$$$$29\!\cdots\!20$$$$T^{10} +$$$$43\!\cdots\!92$$$$T^{11} +$$$$29\!\cdots\!03$$$$T^{12} +$$$$21\!\cdots\!88$$$$T^{13} -$$$$70\!\cdots\!20$$$$T^{14} -$$$$72\!\cdots\!54$$$$T^{15} +$$$$11\!\cdots\!44$$$$T^{16} -$$$$17\!\cdots\!04$$$$T^{17} -$$$$91\!\cdots\!83$$$$T^{18} +$$$$11\!\cdots\!16$$$$T^{19} +$$$$21\!\cdots\!80$$$$T^{20} -$$$$15\!\cdots\!94$$$$T^{21} -$$$$59\!\cdots\!16$$$$T^{22} +$$$$52\!\cdots\!28$$$$T^{23} +$$$$20\!\cdots\!21$$$$T^{24}$$
$83$ $$1 + 4464 T + 7776076 T^{2} + 5918441448 T^{3} + 358006292308 T^{4} - 2898431576154492 T^{5} - 2332832226440682945 T^{6} - 81475410547409243304 T^{7} +$$$$19\!\cdots\!76$$$$T^{8} +$$$$21\!\cdots\!98$$$$T^{9} +$$$$67\!\cdots\!60$$$$T^{10} -$$$$75\!\cdots\!60$$$$T^{11} -$$$$10\!\cdots\!57$$$$T^{12} -$$$$43\!\cdots\!20$$$$T^{13} +$$$$22\!\cdots\!40$$$$T^{14} +$$$$41\!\cdots\!94$$$$T^{15} +$$$$20\!\cdots\!36$$$$T^{16} -$$$$49\!\cdots\!28$$$$T^{17} -$$$$81\!\cdots\!05$$$$T^{18} -$$$$57\!\cdots\!36$$$$T^{19} +$$$$40\!\cdots\!68$$$$T^{20} +$$$$38\!\cdots\!96$$$$T^{21} +$$$$29\!\cdots\!24$$$$T^{22} +$$$$95\!\cdots\!32$$$$T^{23} +$$$$12\!\cdots\!81$$$$T^{24}$$
$89$ $$( 1 - 158 T + 2359364 T^{2} - 294587366 T^{3} + 2380741274284 T^{4} - 254691947575022 T^{5} + 1709394429694985182 T^{6} -$$$$17\!\cdots\!18$$$$T^{7} +$$$$11\!\cdots\!24$$$$T^{8} -$$$$10\!\cdots\!94$$$$T^{9} +$$$$58\!\cdots\!44$$$$T^{10} -$$$$27\!\cdots\!42$$$$T^{11} +$$$$12\!\cdots\!81$$$$T^{12} )^{2}$$
$97$ $$1 - 1608 T + 2107194 T^{2} - 526452544 T^{3} - 693523220127 T^{4} + 2283715532884784 T^{5} - 1050434593951340620 T^{6} -$$$$60\!\cdots\!08$$$$T^{7} +$$$$29\!\cdots\!96$$$$T^{8} -$$$$26\!\cdots\!56$$$$T^{9} +$$$$10\!\cdots\!10$$$$T^{10} +$$$$16\!\cdots\!80$$$$T^{11} -$$$$23\!\cdots\!12$$$$T^{12} +$$$$15\!\cdots\!40$$$$T^{13} +$$$$90\!\cdots\!90$$$$T^{14} -$$$$20\!\cdots\!52$$$$T^{15} +$$$$20\!\cdots\!36$$$$T^{16} -$$$$38\!\cdots\!44$$$$T^{17} -$$$$60\!\cdots\!80$$$$T^{18} +$$$$12\!\cdots\!48$$$$T^{19} -$$$$33\!\cdots\!87$$$$T^{20} -$$$$23\!\cdots\!72$$$$T^{21} +$$$$84\!\cdots\!06$$$$T^{22} -$$$$58\!\cdots\!16$$$$T^{23} +$$$$33\!\cdots\!21$$$$T^{24}$$